Contents

The physics in question are generally in the classical regime, ℏ→0{\displaystyle \hbar \rightarrow 0}.

Materials are "soft": quantitatively, this implies that all relevant energy scales are of the order of kBT{\displaystyle k_{B}T}.

Condensed matter physics deals with systems composed of O(1023){\displaystyle O\left(10^{23}\right)} particles, and statistical mechanics applies. We are always interested in a reduced description, in terms of continuum mechanics and elasticity, hydrodynamics, macroscopic electrodynamics and so on.

We begin with an example from Chaikin & Lubensky, the story of an H2O molecule. This molecule is bound together by a chemical bond which is around 20kBT{\displaystyle 20k_{B}T} at room temperature and not easily broken under normal circumstances. What happens when we put ∼1023{\displaystyle \sim 10^{23}} water molecules is a container? First of all,

with such large numbers we can safely discuss phases of matter: namely

Gas is typical to low density, high temperature and low pressure. It is generally prone to changes in shape and volume, homogeneous, isotropic, weakly interacting and insulating. This is the least ordered form of matter relevant to our scenario, and relatively easy to treat since order parameters are small. The liquid phase is typical of intermediate temperatures. It flows but is not very compressible. It is homogeneous, isotropic, dense and strongly interacting. Its response to external forces depends on the rate of its deformation. Liquids are hard to treat theoretically, as their intermediate properties make simple approximations less effective. The solid is a dense ordered phase with low entropy and strong interactions. It is anisotropic and does not flow, it strongly resists compression and its response to forces depends on the amount of deformation they cause (elastic). Transitions between these phases occur at specific values of thermodynamic parameters (see diagram (1)). First order changes (volume/density "jumps" at the transition, and no jump in pressure/temperature) occur on the lines; at the critical liquid/gas point, second order phase transitions occur; at the triple point, all three phases (solid/liquid/gas) coexist. The systems we are interested in are characterized by several kinds of interactions between their constituent molecules: for example, Coulombic interactions of the form q2r2{\displaystyle {\frac {q^{2}}{r^{2}}}} when charged particles are present, fixed dipole interaction of the form p1⋅p2r3{\displaystyle {\frac {\mathbf {p} _{1}\cdot \mathbf {p} _{2}}{r^{3}}}} when permanent dipoles exist, and almost always induced dipole/van der Waals interaction of the form Δp1⋅Δp2r6{\displaystyle {\frac {\Delta \mathbf {p} _{1}\cdot \Delta \mathbf {p} _{2}}{r^{6}}}}. At close range we also have the "hard core" or steric repulsion, sometimes modeled by a ∼1r12{\displaystyle \sim {\frac {1}{r^{12}}}} potential. Simulations often use the so-called 12−6{\displaystyle 12-6} Lennard-Jones potential U=4ε[(σr)12−(σr)6]{\displaystyle U=4\varepsilon \left[\left({\frac {\sigma }{r}}\right)^{12}-\left({\frac {\sigma }{r}}\right)^{6}\right]}(as pictured in (2)), which with appropriate parameters correctly describes both condensation and crystallization in some cases.

Sidenote

When only the repulsive potential exists (for instance, for billiard balls), crystallization still takes place but no condensation/evaporation phase transition between the liquid and gas phases exists.

Starting from a classical Hamiltonian such as H=∑i(pi22m+VVdW){\displaystyle H=\sum _{i}\left({\frac {\mathbf {p} _{i}^{2}}{2m}}+V_{VdW}\right)}, we can predict all three phases of matter and the transitions between them. In biological systems, this simple picture does not suffice: the basic consideration behind this is that of effects which occur at different scales between the nanometric scale, through the mesoscopic and up to the macroscopic scale. Biological systems are mesoscopic in nature, and their properties cannot be described correctly when a coarse-graining is performed without accurately accounting for mesoscopic properties. A few examples follow:

The most basic assumption we need in order to model liquid crystals is that isotropy at the molecular level is broken: molecules are represented by rods rather than spheres. Such a description was suggested by Onsager and others, and leads to three phases as shown in (3).

This kind of substance is approximately 95% agent, with the remainder water - yet it behaves like a weak solid as long as its deformations are small. This is because a tight formation of ordered cells separated by thin liquid films is formed, and in order for the material to change shape the cells must be rearranged. This need for restructuring is the cause of such systems' solid-like resistance to change.

Interfaces between fluids have interesting properties: they act as a 2D liquid within the interface, yet respond elastically to any bending of the surface. Surfactant molecules will spontaneously form membranes within the same fluid, which also have these properties at appropriate temperatures. Surfactants in solution also form lamellar structures - multilayered structures in which the basic units are the membranes rather than single molecules. 03/19/2009

Natural polymers like rubber have been known since the dawn of history, but not understood. The first artificial polymer was made ∼1905{\displaystyle \sim 1905}. Stadinger was the first to understand that polymers are formed by molecular chains and is considered to be the father of synthetic polymers. Most polymers were made by petrochemical industry. Nylon was born in 1940. Various uses and unique properties (light, strong, thermally insulating; available in many different forms from strings and sheets to bulk; cheap, easy to process, shape and mass-produce...) have made them very attractive commercially. Later on, some leading scientists were Kuhn and Flory in chemistry (30's to 70's) and Stockmayer in physical chemistry (50's and 60's). The famous modern theory of polymers was first formulated by P.G. de Gennes and Sam Edwards.

More generally, this kind of structure is called a homopolymer . Heteropolymers - which have several repeating constituent units - also exist. These can have a random structure (A−B−B−A−B−A...{\displaystyle A-B-B-A-B-A...}) or a block structure ([A]n[B]m[C]l{\displaystyle \left[A\right]_{n}\left[B\right]_{m}\left[C\right]_{l}}), in which case they are called block copolymers . These can self-assemble into complex ordered structures and are often very useful.

Sidenote

For an example, look up ester monomers and polyester, or polyethylene.

Polymerization is also the name of the process by which polymers are synthesized, which involves a chain reaction where a reactive site exists at the end of the chain. Some chemical reactions increase the chain length by one unit, while simultaneously moving the reactive

Consider the example of hydrocarbon polymers, where we have a monomer which is C2H4{\displaystyle \mathrm {C_{2}H_{4}} }(Check this...). As a larger number of such units is joined together to become polyethylene molecules, the material composed of these molecules changes drastically in nature:

The simplest model of an ideal polymer chain is the freely jointed chain (FJC), where each monomer performs a completely independent random rotation. Here, at equilibrium the end-to-end length of the chain is R0≃N12ℓ=L12ℓ12{\displaystyle R_{0}\simeq N^{\frac {1}{2}}\ell =L^{\frac {1}{2}}\ell ^{\frac {1}{2}}}, where L=Nℓ{\displaystyle L=N\ell } is the contour length.

A slightly more realistic model is the freely rotating chain

(FRC), where monomers are locked at some chemically meaningful bond angleϑ{\displaystyle \vartheta } and rotate freely around it via the torsional angleφ{\displaystyle \varphi }. Here,

Note that for ⟨cos⁡ϑ⟩=0{\displaystyle \left\langle \cos \vartheta \right\rangle =0} we find that ℓeff=ℓ{\displaystyle \ell _{eff}=\ell } and this is identical to the FJC. For very small ϑ∼ε{\displaystyle \vartheta \sim \varepsilon }, we can expand the cosine an obtain

A second possible improvement is the hindered rotation (HR) model. Here the angles φi{\displaystyle \varphi _{i}} have a minimum-energy value, and are taken from an uncorrelated Boltzmann distribution with some

Another option is called the rotational isomeric state model. Here, a finite number of angles are possible for each monomer junction and the state of the full chain is given in terms of these. Correlations are also taken into account and the solution is numeric, but aside from a complicated ℓeff{\displaystyle \ell _{eff}} this is still an ideal chain with R02≃Lℓeff∼N{\displaystyle R_{0}^{2}\simeq L\ell _{eff}\sim N}.

For the polymer chain of (5), obviously we will always have ⟨RN⟩=0{\displaystyle \left\langle \mathbf {R} _{N}\right\rangle =0}. The variance, however, is generally not zero: using RN=∑i=1Nri{\displaystyle \mathbf {R} _{N}=\sum _{i=1}^{N}\mathbf {r} _{i}},

and since the φi{\displaystyle \varphi _{i}} are independent and any averaging over a sine of cosine of one or more of them will result in a zero, only the φi{\displaystyle \varphi _{i}} independent terms survive and by recursion this correlation has the simple form

At large N{\displaystyle N} we can approximate the two sums in k{\displaystyle k} by the series ∑k=1∞cosk⁡ϑ=cos⁡ϑ1−cos⁡ϑ{\displaystyle \sum _{k=1}^{\infty }\cos ^{k}\vartheta ={\frac {\cos \vartheta }{1-\cos \vartheta }}}, giving

To go back from this to the FRC limit, we would consider a chain with a random distribution of ϑ{\displaystyle \vartheta } angles such that ⟨cos⁡ϑ⟩=0{\displaystyle \left\langle \cos \vartheta \right\rangle =0}.

In fact, for N→∞{\displaystyle N\rightarrow \infty } the central limit theorem tells us that x=∑ixi{\displaystyle x=\sum _{i}x_{i}} will have a Gaussian distribution for any distribution of the xi{\displaystyle x_{i}}. This can be extended to d{\displaystyle d} dimensions

We are left with the task of evaluating the integral. This can be done analytically with the Laplace method for large N{\displaystyle N}, since the largest contribution is around kℓ=0{\displaystyle k\ell =0}: we can approximate (sin⁡kℓkℓ)N{\displaystyle \left({\frac {\sin k\ell }{k\ell }}\right)^{N}} by (1−(kℓ)26+...)N≃e−(kℓ)2N6{\displaystyle \left(1-{\frac {\left(k\ell \right)^{2}}{6}}+...\right)^{N}\simeq e^{-{\frac {\left(k\ell \right)^{2}N}{6}}}}.

This is, of course, the same Gaussian form we have obtained from the random walk (we have done the special case of d=3{\displaystyle d=3}, but once again this process can be repeated for a general dimension d≥1{\displaystyle d\geq 1}).

In considering the ϑ→0{\displaystyle \vartheta \rightarrow 0} limit of the freely rotating chain, we have seen that ℓeff∼ℓϑ2→∞{\displaystyle \ell _{eff}\sim {\frac {\ell }{\vartheta ^{2}}}\rightarrow \infty }. This is of course unphysical, and this limit is actually important for many interesting cases of stiff chains (for instance, DNA). If we take the N→∞{\displaystyle N\rightarrow \infty } limit along with ϑ→0{\displaystyle \vartheta \rightarrow 0}

The fact that R0∼N{\displaystyle R_{0}\sim N} rather than R0∼N12{\displaystyle R_{0}\sim N^{\frac {1}{2}}} is a result of the long-range correlations we have introduced, and is an indication that at this regime the material is in an essentially different phase. Somewhere between the ideal chain and the rigid rod, a crossover regime must exist.

Sidenote

While an ideal chain has R0∼N12{\displaystyle \scriptstyle R_{0}\sim N^{\frac {1}{2}}} and a rigid rod has R0∼N{\displaystyle \scriptstyle R_{0}\sim N}, in general polymer chains can have a scaling law R0∼Nν{\displaystyle \scriptstyle R_{0}\sim N^{\nu }}. The power ν{\displaystyle \scriptstyle \nu } need not be an integer.

For ℓp≪Rmax{\displaystyle \ell _{p}\ll R_{max}} we can neglect the exponent, obtaining

This therefore returns us to the ideal chain limit, with a Kuhn length ℓeff=2ℓp{\displaystyle \ell _{eff}=2\ell _{p}}. The crossover phenomenon we discussed occurs on the chain itself here as we observe correlation between its pieces at differing length scales: at small scales (∼ℓp{\displaystyle \sim \ell _{p}}) it behaves like a rigid rod, while at long scales we have an uncorrelated random walk. An interesting example is a DNA chain, which can be described by a worm-like chain with ℓp≈500Å{\displaystyle \ell _{p}\approx 500\mathrm {\AA} } and Rmax≃10μm≫ℓp{\displaystyle R_{max}\simeq 10\mu m\gg \ell _{p}}: it will therefore typically cover a radius of R0∼7000Å{\displaystyle R_{0}\sim 7000\mathrm {\AA} }.

since UN(R)=UN(0){\displaystyle U_{N}\left(\mathbf {R} \right)=U_{N}\left(0\right)} for an ideal chain.

What does FN(R){\displaystyle F_{N}\left(\mathbf {R} \right)} mean? It represents the energy needed to stretch the polymer, and this energy is ∼R2{\displaystyle \sim R^{2}} like a harmonic spring (U∼12kx2{\displaystyle U\sim {\frac {1}{2}}kx^{2}}) with k=3kBTNℓ2∼TN{\displaystyle k={\frac {3k_{B}T}{N\ell ^{2}}}\sim {\frac {T}{N}}}. Note that the polymer becomes less elastic (more rigid) as the temperature increases, unlike most solids. This is a physical result and can be verified experimentally: for instance, the spring constant of rubber (which is made of networks of polymer chains) increases linearly with temperature. Consider an experiment where instead of holding the chain at constant length, we apply a perturbatively weak force ±f{\displaystyle \pm \mathbf {f} } to its ends and measure its average length. We can perform a Legendre transform between distance and force: from equality of forces along the direction

Numerically, with a nanometric ℓ{\displaystyle \ell } and at room temperature the forces should be in the picoNewton range to meet this requirement. A more rigorous treatment which works at arbitrary forces can be carried out by considering an FJC with oppositely charged (±q{\displaystyle \pm q}) ends in an electric field E∥z^{\displaystyle \mathbf {E} \parallel \mathbf {\hat {z}} }. The chain's sites are at ri{\displaystyle \mathbf {r} _{i}} with R≡RN−R0{\displaystyle \mathbf {R} \equiv \mathbf {R} _{N}-\mathbf {R} _{0}}.

A fractal is an object with fractaldimensionality , called also the Hausdorff dimension . This implies a new definition of dimensionality, which we will discuss. Consider a sphere of radius R{\displaystyle R}. It is considered three-dimensional because it has V=4π3R3{\displaystyle V={\frac {4\pi }{3}}R^{3}} and M=ρV∼RD{\displaystyle M=\rho V\sim R^{D}} for D=3{\displaystyle D=3}. A plane has by the same reasoning M∼RD{\displaystyle M\sim R^{D}} for D=2{\displaystyle D=2}, and is therefore a 2D{\displaystyle 2D} object. Fractals are mathematical objects such that by the same sort of calculation they will have M∼RDf{\displaystyle M\sim R^{D_{f}}}, for a Df{\displaystyle D_{f}} which is not necessarily an integer number (this definition is due to Hausdorff). One example is the Koch curve (see (7)): in each of its iterations, we decrease the length of a segment by a factor

Note that a fractal's "real" length is infinite, and its approximations will depend on the resolution. The structure exhibits self-similarity: namely, on different length scales it will look the same. This can be seen in the Koch snowflake: at any magnification, a part of the curve looks similar to the whole curve. There's a very nice animation of this in Wikipedia. The total length of the curve depends on the the ruler used to measure it: the actual length at iteration n{\displaystyle n} is L0(43)n{\displaystyle L_{0}\left({\frac {4}{3}}\right)^{n}}.

Consider the ideal Gaussian chain again. It has R02=Nℓ2∼N{\displaystyle R_{0}^{2}=N\ell ^{2}\sim N}. Since N{\displaystyle N} is proportional to the mass, we have an object with a fractal dimension of 2 no matter what the dimensionality of the actual space is. We can say that a polymer in d{\displaystyle d}-space fills only Df≤d{\displaystyle D_{f}\leq d} dimensions of the space it occupies, where Df{\displaystyle D_{f}} is 2 for an ideal polymer Gaussian and 2≤Df≤d{\displaystyle 2\leq D_{f}\leq d} in general. Flory has shown that in some cases a non-ideal polymer can also have Df<2{\displaystyle D_{f}<2}, in particular when a self-avoiding walk (SAW) is accounted for. The SAW as opposed to the Gaussian walk (GW) is the defining property of a physical rather than ideal polymer, and gives a fractal dimension of Df≈1.66{\displaystyle D_{f}\approx 1.66}. A collapsed polymer has Df=3{\displaystyle D_{f}=3} and fills space completely. Note that two polymers with fractal dimensions Df{\displaystyle D_{f}} and Df∗{\displaystyle D_{f}^{*}} do not "feel" each other statistically if Df+Df∗<d{\displaystyle D_{f}+D_{f}^{*}<d}.

This model is also known as LGC. We start from an FJC in 3D where Ψ=∏iψ(ri){\displaystyle \Psi =\prod _{i}\psi \left(\mathbf {r} _{i}\right)} and ψ(ri)=14πℓ2δ(ri−ℓ){\displaystyle \psi \left(\mathbf {r} _{i}\right)={\frac {1}{4\pi \ell ^{2}}}\delta \left(\mathbf {r} _{i}-\ell \right)}. By the central limit theorem R=∑iri{\displaystyle \mathbf {R} =\sum _{i}\mathbf {r} _{i}} will always be taken from a Gaussian distribution when the number of monomers is large (whatever the form of ψ{\displaystyle \psi }, as long as it

is symmetrical around zero such that ⟨ri⟩=0{\displaystyle \left\langle \mathbf {r} _{i}\right\rangle =0}):

An exact property of the Gaussian distributions we have been using is that a sub chain of m−n{\displaystyle m-n} monomers (such as the sub chain starting at index m{\displaystyle m} and ending at n{\displaystyle n}) will also have a a Gaussian distribution

At the continuum limit, we will get Wiener distributions : the correct way to calculate the limit is to take N→∞{\displaystyle N\rightarrow \infty } and ℓ→0{\displaystyle \ell \rightarrow 0} with Nℓ=L{\displaystyle N\ell =L} remaining constant. The length along the chain up to site n{\displaystyle n} is then described by nℓ→s{\displaystyle n\ell \rightarrow s}, 0≤s≤L{\displaystyle 0\leq s\leq L}. At this limit we can also substitute derivatives ∂R∂s=1ℓ∂R∂n{\displaystyle {\frac {\partial \mathbf {R} }{\partial s}}={\frac {1}{\ell }}{\frac {\partial \mathbf {R} }{\partial n}}} for the finite differences Ri−Ri−1ℓ{\displaystyle {\frac {\mathbf {R} _{i}-\mathbf {R} _{i-1}}{\ell }}},

Consider what happens when we hold the ends of a chain defined by {Ri}{\displaystyle \left\{\mathbf {R} _{i}\right\}} in place, such that R0=R′{\displaystyle \mathbf {R} _{0}=\mathbf {R} ^{\prime }} and RN=R{\displaystyle \mathbf {R} _{N}=\mathbf {R} }. We can calculate the probability

At the continuum limit the definition of the chain configurations translates into a function R(n){\displaystyle \mathbf {R} \left(n\right)} and the product of integrals can be taken as a path integral according to ∏i=1N−1∫dRi→∫DR(n){\displaystyle \prod _{i=1}^{N-1}\int \mathrm {d} \mathbf {R} _{i}\rightarrow \int {\mathcal {D}}\mathbf {R} \left(n\right)}. The probability for each configuration with our constraint is a functional

We now introduce the Green's function G(R,R′;N),{\displaystyle G\left(\mathbf {R} ,\mathbf {R} ^{\prime };N\right),}which as we will soon see describes the evolution from R′{\displaystyle \mathbf {R} ^{\prime }}

to R{\displaystyle \mathbf {R} } in N{\displaystyle N} steps. We define it as:

Note that while the nominator is proportional to the probability PN{\displaystyle P_{N}}, the denominator does not include include the external potential.

G{\displaystyle G} has several important properties:

It is equal to the exact probability PN{\displaystyle P_{N}} for Gaussian chains in the absence of external potential.

If we consider that the chain might be divided into one sub chain between step 0{\displaystyle 0} and i{\displaystyle i} and a second sub chain from step i{\displaystyle i} to step N{\displaystyle N}, then

The Green's function is defined as 0 for N<0{\displaystyle N<0} and is equal to δ(R−R′){\displaystyle \delta \left(\mathbf {R} -\mathbf {R} ^{\prime }\right)} when N→0{\displaystyle N\rightarrow 0} in order to satisfy the boundary conditions.

where the φk{\displaystyle \varphi _{k}} are solutions of the homogeneous equation (L−En)φn=0{\displaystyle \left({\mathcal {L}}-E_{n}\right)\varphi _{n}=0}.

Example A polymer chain in a box of dimensions Lx×Ly×Lz{\displaystyle L_{x}\times L_{y}\times L_{z}}: The potential U{\displaystyle U} is 0{\displaystyle 0} within the box and ∞{\displaystyle \infty } on the edges. The boundary conditions are G(R,R′;N)=0{\displaystyle G\left(\mathbf {R} ,\mathbf {R} ^{\prime };N\right)=0} if R{\displaystyle \mathbf {R} } or R′{\displaystyle \mathbf {R} ^{\prime }} are on the boundary. The

This is equivalent to a dilute gas of polymers (done here for a single chain). At the opposite limit, Li≪Nℓ{\displaystyle L_{i}\ll {\sqrt {N}}\ell }, the polymer should be "squeezed". The Gaussian approximation will be no good if we squeeze too hard, but at least for some intermediate regime

There is a large extra pressure caused by the "squeezing" of the chain and the corresponding loss of its entropy.

04/30/2009

The same formalism can be used to treat polymers near a wall or in a well near a wall, for instance (see the homework for details). In the well case, like in the similar quantum problem, we will have bound states for T<Tc{\displaystyle T<T_{c}} (where the critical temperature is defined by a critical value of βcV0=V0kBTc{\displaystyle \beta _{c}V_{0}={\frac {V_{0}}{k_{B}T_{c}}}}, and describes the condition for the potential well to be "deep" enough to contain a bound state).

where N{\displaystyle N} is positive and the En{\displaystyle E_{n}} are real and ordered (assuming no degeneracy, E0<E1<E2<...{\displaystyle E_{0}<E_{1}<E_{2}<...}), at large N{\displaystyle N} we can neglect

This is possible because the exponent is decreasing rather than oscillating, as it is in the quantum mechanics case. Taking only the first term in this series is called the dominant ground state approximation .

So far, in treating Gaussian chains, we have neglected any long-ranged interactions. However, polymers in solution cannot self-intersect, and this introduces interactions V(Ri−Rj){\displaystyle V\left(\mathbf {R} _{i}-\mathbf {R} _{j}\right)} into the picture which are local in real-space, but are long ranged in terms of the contour spacing - that is, they are not limited to i≈j{\displaystyle i\approx j}. The importance of this effect depends on dimensionality: it is easy to imagine that intersections in 2D are more effective in restricting a polymer's shape than intersections in 3D.

The interaction potential V(r){\displaystyle V\left(\mathbf {r} \right)} can in general have both attractive and repulsive parts, and depends on the detailed properties of the solvent. If we consider it to be due to a long ranged attractive Van der-Waals interaction and a short ranged repulsive hard-core interaction, it might be modeled by a 6−12{\displaystyle 6-12} Lennard-Jones potential. To treat interaction perturbatively within statistical mechanics, we can use a virial expansion (this is a statistical-mechanical expansion in powers of the density, useful for systematic perturbative corrections to non-interacting calculations when one wants to include

This can be positive (signifying net repulsion between the particles) at kBT>εln⁡87{\displaystyle k_{B}T>{\frac {\varepsilon }{\ln {\frac {8}{7}}}}} or negative (signifying attraction) for kBT<εln⁡87{\displaystyle k_{B}T<{\frac {\varepsilon }{\ln {\frac {8}{7}}}}}. While the details of this calculation depend on our choice and parametrization of the potential, in general we will have some special temperature known as the ϑ{\displaystyle \vartheta } temperature (in our case kBϑ=εln⁡87{\displaystyle k_{B}\vartheta ={\frac {\varepsilon }{\ln {\frac {8}{7}}}}})

where

v2(ϑ)=0.{\displaystyle v_{2}\left(\vartheta \right)=0.}

This allows us to define a good solvent: such a solvent must have T>ϑ{\displaystyle T>\vartheta } at our working temperature. This assures us (within the second Virial approximation, at least) that the interactions are repulsive and (as can be shown separately) the chain is swollen . A bad solvent for which T<ϑ{\displaystyle T<\vartheta } will have attractive interactions, resulting in collapse . A solvent for which T=ϑ{\displaystyle T=\vartheta } is called a ϑ{\displaystyle \vartheta } solvent, and returns us to a Gaussian chain unless the next Virial coefficient is taken.

A common numerical treatment for this kind of system is to draw the polymer on a grid and make Monte-Carlo runs, where steps must be self-avoiding and their probability is taken from a thermal distribution while maintaining detailed balance. This gives in 3D RN≃ℓNν{\displaystyle R_{N}\simeq \ell N^{\nu }} where ν≈0.588{\displaystyle \nu \approx 0.588}.

A connection between SAWs and critical phenomena was made by de Gennes in the 1970's. Some of the similarities are summarized in the table below. Using renormalization group methods, de Gennes showed by analogy

This gives in 3D a result very close to the SAW: νRG=12+116+15512+ϑ(ε3)=0.5625+ϑ(ε3){\displaystyle \nu _{RG}={\frac {1}{2}}+{\frac {1}{16}}+{\frac {15}{512}}+\vartheta \left(\varepsilon ^{3}\right)=0.5625+\vartheta \left(\varepsilon ^{3}\right)}.

This is a very crude model which gives surprisingly good results. We write the free energy as Ftot(R)=Fint+Fent{\displaystyle F_{tot}\left(R\right)=F_{int}+F_{ent}}. For the entropic part we take the expression for an ideal chain: SN(R)=−d2kBR2Nℓ2+SN(0){\displaystyle S_{N}\left(R\right)=-{\frac {d}{2}}k_{B}{\frac {R^{2}}{N\ell ^{2}}}+S_{N}\left(0\right)}, Fent=−TSN{\displaystyle F_{ent}=-TS_{N}}. For the interaction, we use the second virial

This exponent is exact for 1, 2 and 4 dimensions, and gives a very good approximation (0.6) for 3 dimensions, but it misses completely for more than 4 dimensions. For a numerical example consider a polymer of ∼105{\displaystyle \sim 10^{5}} monomers each of which is about 5Å{\displaystyle 5\mathrm {\AA} } in length.

This difference is large enough to be experimentally detectable by the scattering techniques to be explained next.

The reason the Flory method provides such good results turns out to be a matter of lucky cancellation between two mistakes, both of which are by orders of magnitude: the entropy is overestimated and the correlations are underestimated. This is discussed in detail in all the books.

The seminal article of S.F. Edwards in 1965 was the first application of field-theoretic methods to the physics of polymers. To insert interactions into the Wiener distribution, we take sum over the two-body interactions 12∑ijV(Ri−Rj){\displaystyle {\frac {1}{2}}\sum _{ij}V\left(\mathbf {R} _{i}-\mathbf {R} _{j}\right)} to the continuum limit 12∫0Ndn∫0NdmV(R(m)−R(n)){\displaystyle {\frac {1}{2}}\int _{0}^{N}\mathrm {d} n\int _{0}^{N}\mathrm {d} mV\left(\mathbf {R} \left(m\right)-\mathbf {R} \left(n\right)\right)}.

This formalism is rather complicated and not much can be done by hand. One possible simplification is to consider an excluded-volume (or self-exclusion) interaction of Dirac delta function form, which prevents

With expressions of this sort, one can apply standard field-theory/many-body methods to evaluate the Green's function and calculate observables. This is more advanced and we will not be going into it. 05/07/2009

Materials can be probed by scattering experiments, and for dilute polymer solutions this is one way to learn about the polymers within them. Laser scattering requires relatively little equipment and can be done in any lab, while x-ray scattering (SAXS) requires a synchrotron and neutron scattering (SANS) requires a nuclear reactor. We will discuss structural properties on the scale of chains rather than individual monomers, which means relatively small wavenumbers. It will also soon be clear that small angles are of interest.

Sidenote

Modeling the monomers as points is reasonable when considering probing on the scale of the complete chain.

If we assume that the individual monomers act as point scatterers (see (8)) and consider a process which scatters the incoming wave at ki{\displaystyle \mathbf {k} _{i}} to kf{\displaystyle \mathbf {k} _{f}}, we can define a scattering angle ϑ{\displaystyle \vartheta } and a scattering wave vector k=kf−ki{\displaystyle \mathbf {k} =\mathbf {k} _{f}-\mathbf {k} _{i}} (which becomes smaller in magnitude as the angle ϑ{\displaystyle \vartheta } becomes smaller). We then measure scattered waves at some outgoing angle for some incoming angle as illustrated in (9), where in fact many chain scatterers are involved we should have an ensemble average over the chain configurations (which should be incoherent since the chains are far apart compared with the typical decoherence length scale). All this is discussed in more detail below.

Sidenote

For this kind of experiment to work with lasers or x-rays, there must be a contrast : the polymer and solvent must have different indices of refraction. X-Ray experiments rely on different electronic densities. In neutron scattering experiments, contrast is achieved artificially by labeling the polymers or solvent - that is, replacing hydrogen with deuterium.

Within a chain scattering is mostly coherent such that that the scattered wavefunction is Ψ=∑i=1Naieik⋅Ri{\displaystyle \Psi =\sum _{i=1}^{N}a_{i}e^{i\mathbf {k} \cdot \mathbf {R} _{i}}}. The intensity or power should be proportional to I=|Ψ|2=∑i,j=1Naiaj∗eik⋅(Ri−Rj){\displaystyle I=\left|\Psi \right|^{2}=\sum _{i,j=1}^{N}a_{i}a_{j}^{*}e^{i\mathbf {k} \cdot \left(\mathbf {R} _{i}-\mathbf {R} _{j}\right)}}).

If we specialize to homogeneous chains where ai=a{\displaystyle a_{i}=a}, then

This expression is suitable for a single static chain in a specific configuration {Ri}{\displaystyle \left\{\mathbf {R} _{i}\right\}}. For an ensemble of chains in solution, we average over all chain configurations incoherently,

The normalization is with respect to the unscattered wave at k=0{\displaystyle k=0}, |Ψ(0)|2=a2N2{\displaystyle \left|\Psi \left(0\right)\right|^{2}=a^{2}N^{2}}. Note that in an isotropic system like the system of chain molecules in a solvent, the structure factor must depend only on the magnitude of k{\displaystyle k}.

Inserting the expression for Ψ2{\displaystyle \Psi ^{2}} into the above equation gives

From an experimental point of view, we can plot S{\displaystyle S} as a function of k2∼sin2⁡ϑ2{\displaystyle k^{2}\sim \sin ^{2}{\frac {\vartheta }{2}}} and determine the polymer's gyration radius Rg{\displaystyle R_{g}} from the slope.

The approximation we have made is good when kRg∼sin⁡ϑ2λRg≪1{\displaystyle kR_{g}\sim {\frac {\sin {\frac {\vartheta }{2}}}{\lambda }}R_{g}\ll 1}, and this determines the range of angles that should be taken into account: we must have sin⁡ϑ2∼ϑ2≲λRg{\displaystyle \sin {\frac {\vartheta }{2}}\sim {\frac {\vartheta }{2}}\lesssim {\frac {\lambda }{R_{g}}}}. For laser scattering usually λ∼500nm{\displaystyle \lambda \sim 500\mathrm {nm} } (about enough to measure Rg{\displaystyle R_{g}}) while for neutron scattering λ∼0.3nm{\displaystyle \lambda \sim 0.3\mathrm {nm} } (meaning we must take only very small angles into account to measure Rg{\displaystyle R_{g}}, but also allowing for more detailed information about correlations within the chain to be collected).